#138635

order	512
nilpotency class	4
isoclinism	isoclinic class of #138616
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

368 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(2,5)(3,6)(7,8)(9,10,15,22,16,23,39,54,17,24,14,21,38,53,13,20)(11,25,28,47,29,52,27,57,30,55,37,18,42,45,40,46)(12,31,34,58,35,56,32,19,36,49,33,50,43,48,41,51)(26,60,62,67,63,44,68,69,64,70,65,72,66,71,61,59);(1,2,6,8,4,7,3,5)(9,11,16,42,17,30,38,29)(10,18,23,57,24,47,53,46)(12,26,43,63,36,64,35,66)(13,37,15,27,39,28,14,40)(19,44,58,70,51,71,50,60)(20,52,22,25,54,45,21,55)(31,67,48,69,49,72,56,59)(32,62,34,68,41,65,33,61);(9,12,17,36)(10,19,24,51)(11,26,30,64)(13,34,39,33)(14,32,15,41)(16,43,38,35)(18,44,47,71)(20,49,54,31)(21,48,22,56)(23,58,53,50)(25,59,55,69)(27,65,40,62)(28,61,37,68)(29,66,42,63)(45,67,52,72)(46,60,57,70)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=i,d^2=ghi,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=ehi,f^b=fh,h^b=hi,d^c=dhi,e^c=ehi,f^c=fhi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45831);